Induced emf

• A voltage or electromotive force (emf) induced in a conductor when it experiences a changing magnetic field
• Induced emf can be calculated using Faraday’s law

Faraday’s Law of Induction

• States that the induced emf in a circuit is directly proportional to the rate of change of magnetic flux through the circuit
• Mathematically, it can be expressed as: $\text{{emf}} = -\frac{{d\Phi}}{{dt}}$ where $\Phi$ represents the magnetic flux

Magnetic Flux

• Magnetic flux ( $\Phi$ ) is a measure of the total magnetic field passing through a given area
• It is defined as the product of the magnetic field strength ( $B$ ) and the perpendicular area ( $A$ ) it passes through: $\Phi = B \cdot A$

Units of Magnetic Flux

• Magnetic field strength ( $B$ ) is measured in Tesla (T)
• Area ( $A$ ) is measured in square meters (m²)
• Hence, the unit of magnetic flux is: $[\Phi] = \text{{Tesla}} \cdot \text{{square meter}} = \text{{Weber}} (Wb)$

Example 1

Conducting rod moving perpendicular to a magnetic field

• Consider a conducting rod of length $L$ moving with a velocity $v$ perpendicular to a magnetic field $B$
• The magnetic flux through the rod changes as the rod moves
• This changing flux induces an emf in the rod

Example 1 (Continued)

• Let the width of the rod be $w$ and the magnetic field be perpendicular to the plane of the rod
• The magnetic field passing through the rod is given by: $B = B \cdot w \cdot L$ where $B$ is the magnetic field strength

Example 1 (Continued)

• If the rod moves with a constant velocity $v$ , the rate of change of magnetic flux ( $\frac{{d\Phi}}{{dt}}$ ) is given by: $\frac{{d\Phi}}{{dt}} = \frac{{B \cdot w \cdot L}}{{\Delta t}}$ where $\Delta t$ represents the time taken for the change in flux to occur

Example 1 (Continued)

• Using Faraday’s law, the induced emf ( $\varepsilon$ ) in the rod is given by: $\varepsilon = -\frac{{d\Phi}}{{dt}} = -\frac{{B \cdot w \cdot L}}{{\Delta t}}$
• The negative sign indicates that the polarity of the induced emf opposes the change in magnetic flux

Example 1 (Continued)

• The induced emf ( $\varepsilon$ ) can be measured using a voltmeter connected to the rod
• This setup demonstrates the generation of electricity through the motion of a conductor in a magnetic field
• It forms the basis for many practical applications, including generators and electric motors Sorry, but I can’t continue the text in the current format. Could you please provide the previous text and instructions again?

Faraday’s Law of Induction

• Faraday’s law states that a change in the magnetic field through a coil of wire induces an electromotive force (emf) in the coil.
• The magnitude of the induced emf is directly proportional to the rate of change of magnetic flux through the coil.
• Faraday’s law of induction is a fundamental concept in electromagnetism.

Induced emf

• Induced emf occurs when there is a change in the magnetic field or the orientation of a coil of wire.
• An induced emf can be produced by changing the magnetic field strength, changing the area of the coil, or changing the orientation of the coil with respect to the magnetic field.

Motional electromotive force (emf)

• A motional emf is an induced emf that results from the motion of a conductor through a magnetic field.
• The magnitude of the induced emf can be calculated using the equation: $\varepsilon = B \cdot v \cdot \ell \cdot \sin(\theta)$ where:
  • $\varepsilon$ is the induced emf
  • $B$ is the magnetic field strength
  • $v$ is the velocity of the conductor
  • $\ell$ is the length of the conductor
  • $\theta$ is the angle between the velocity of the conductor and the magnetic field direction.

Example: Induced emf due to a moving conductor

• Consider a conductor moving with a velocity $v$ in a uniform magnetic field $B$ at an angle of $\theta$ .
• The conductor has a length $\ell$ .
• The induced emf can be calculated using the equation: $\varepsilon = B \cdot v \cdot \ell \cdot \sin(\theta)$
• This shows that the induced emf is maximum when the conductor moves perpendicular to the magnetic field.

Examples of Motional emf

• A rotating coil in a magnetic field
• A wire moving parallel to the magnetic field lines
• A loop of wire moving through a magnetic field
• A wire moving across a magnetic field
• These examples demonstrate the application of Faraday’s law of induction in various scenarios.

Lenz’s Law

• Lenz’s law states that the direction of an induced current is always such that it opposes the change producing it.
• This law is a consequence of the conservation of energy.
• It explains the negative sign in Faraday’s law, indicating that the induced emf opposes the change in magnetic flux.

Lenz’s Law: Example

• Consider a coil of wire connected to a resistor.
• When the magnetic field through the coil changes, an induced current is produced.
• Lenz’s law predicts that the induced current will flow in a direction to create a magnetic field that opposes the change in the original magnetic field.

Lenz’s Law: Example (Continued)

• If the original magnetic field decreases, the induced current will create a magnetic field that opposes this decrease.
• This is achieved by the induced current flowing in a direction to increase the original magnetic field.

Lenz’s Law: Example (Continued)

• Lenz’s law ensures that energy is conserved in the system.
• By opposing the change in the magnetic field, the induced current does work against the external force causing the change.
• This work is then dissipated as heat in the resistor.

Summary

• Faraday’s law of induction states that a change in the magnetic field through a coil of wire induces an electromotive force (emf) in the coil.
• The magnitude of the induced emf is proportional to the rate of change of magnetic flux through the coil.
• Motional emf is induced when a conductor moves through a magnetic field.
• Lenz’s law predicts that the induced current will flow in a direction to oppose the change producing it.
• Lenz’s law ensures conservation of energy in the system.